Exploring the Solving of Algebraic Equations Through Mental Algebra
For numerous years, mathematics teacher educators have attempted to find ways to enrich their future mathematics teachers’ understandings of mathematics. This has often been done with the intention of deepening and making future teachers more mathematically flexible, particularly in view of their future interactions with students’ own understandings of mathematics. Because doing mental mathematics is well recognized as an occasion for promoting meaning making and for enlarging one’s repertoire of ways of solving (see e.g., Reys & Nohda, 1994; Schoen & Zweng, 1986), it appears as an interesting approach to explore for attaining this precise goal with future mathematics teachers. I therefore report in this short article on a study undertook with future secondary mathematics teachers, where we asked them to solve usual algebraic equations of the form Ax+B=C, Ax+B=Cx+D, Ax/B=C/D, Ax 2+Bx+C=0 without paper or pencil or any other material aids in a restricted period of time (about 15 seconds). The activity was organized in the following way: (1) an equation was offered in writing to the group; (2) future teachers solved the equation mentally; (3) at the teacher educator’s signal they wrote their answer on a piece of paper; (4) answers and strategies were orally shared with the group; (5) the cycle restarted for another equation.
As the activity unfolded, diverse strategies were shared for solving algebraic equations. Through those strategies emerged an interesting variety of meanings (implicitly or not) about what solving an algebraic equation can represent. In what follows, I outline this variety of strategies and meanings, in order to illustrate how doing mental algebra can represent an occasion for the enrichment of future teacher’ mathematical experiences.
Underneath this meaning is the notion of a conditional equality, where it is not only the idea of finding the answers/values that make the equation true, but also the fact that the equality can be true or untrue.
When future teachers were given 5x+6+4x+3=–1+9x to solve, some rapidly asserted that there was no solution, because one can rapidly see 9x on both sides of the equation as well as the fact that the remaining numbers on each sides do not equate. It thus led to the conclusion that there was no number that could satisfy this given equation, since no x, whatever it could be, could succeed in making different numbers equal. This strategy is related to what is often termed “global reading” of the equation (Bednarz & Janvier, 1992), that requires consideration of the equation as a whole prior to entering in algebraic manipulations, or what Arcavi (1994) calls a priori inspection of symbols, which is a sensitivity to analyze algebraic expressions before making a decision about their solution 1.
Another strategy future teachers engaged in was one of “solving followed by validation”. When having to solve x 2–4=5, one future teacher rapidly transformed it into x 2=9, obtaining 3 as an answer. However, because he knows being in a mental mathematics context and is aware that his answers in this context are often rapidly enunciated and can lack precision, he decided to verify his answer. By mentally verifying if (3) 2=9, he realized that (–3) 2 also gives 9 and then readjusted his solution. This manner of solving the equation gets close to the idea not only of finding one value that makes the equation true, but also of finding all values that make it true.
This meaning requires reading the equation as a series of operations applied to a number (here x) and attempting to undo these operations to find that number.
When having to solve equations like x 2–4=5, future teachers would say: “My number was squared and then 4 was taken away, thus I need to add 4 and take the square root”. Or, for 4x+2=10, “What is my number which after having multiplied by 4 and added 2 to it gives me 10?” These are similar to inverse methods of solving found in Filloy and Rojano (1989) or of Nathan and Koedinger’s (2000) “unwinding” approach, where operations are arithmetically “undone” to arrive at a value for x. As Filloy and Rojano explain, when using this method “it is not necessary to operate on or with the unknown” (p. 20), as it becomes a series of arithmetical operations performed on numbers. In this particular case, solving the algebraic equation is focused on finding a way to arrive at isolating x, in an arithmetic context.
This meaning focuses on the idea that is often called “the balance” principle, where one operates identically on both sides of the equation to maintain the equality and obtain “x=something”. For example, when solving 2x+3=5, students would subtract 3 on each side and then divide by 2.
This one is about seeing each side of the equality as representing two functions, and thus attempting to solve them as a system of equations to find intersecting points, if any. For example, when solving x 2–4=5, some future teachers attempted to depict the equation as the comparison of two equations in a system of equations (y=x 2–4 and y=5) and finding the intersecting point of those two equations in the graph. To do so, one future teacher represented the line y=5 in the graph and then also positioned y=x 2–4. The latter was referred to the quadratic function y=x 2, which crosses y=5 at. In the case of y=x 2–4, the function is translated of 4 downwards in the graph, and then the 5 of the line y=5 becomes a 9 in terms of distances. Hence, how does one obtain an image of 9 with the function y=x 2? With an x=±3, where the function y=x 2–4 cuts the line y=5. The following graph (Figure 1) offers an illustration of what was done, mentally, by the future teacher.
Solving an algebraic equation in this case is not about finding the values that make the equation true, but about finding the x that satisfies both equations for the same y, about finding the x coordinate that, for the same y, is part of each function.
Meaning 5: Solving an algebraic equation is … finding the values that nullifies the equation
This meaning focuses on the equal sign as giving an answer (see e.g. Davis, 1975), but where operations are conducted so that all the “information” ends up being on one side of the equation in order to obtain 0 on the other side. The intention then becomes to find the value of x that nullifies that equation, that is, that makes it equal to 0.
One example of a strategy engaged in was again related to seeing the equation in a functional view, but finding the values of x that give a null y-value, or what is commonly called finding the zeros of the function where the function intersects the x-axis when y=0. For x 2–4=5, transformed in x 2–9=0, the future teacher aimed mentally at solving (x+3)(x–3)=0, leading to ±3. The quest was mainly finding the values that nullify the function y=x 2–9, which gave point(s) for which the image of the function was zero. Another way of doing it, less in a function-orientation, is to use “binomial expansion” for seeing that for the product to be null it requires that one of the two factors be null. This said, one needs to use neither a function nor binomial expansion to find what nullifies the equation. For example, if x+4=3 is transformed in x+1=0, one finds that –1 is what makes the left side of the equation equal to 0.
This meaning was engaged with for equation written in fractional form (e.g. Ax/B=C or Ax/B=C/D). In these cases, the equation was conceived as a proportion, where the ratio between numerators and denominators was seen as the same or consistent. The equality here is not seen as conditional but is taken for granted as true, leading at conserving the ratio between numerator and denominator in the proportion.
For example, for , reversed to, future teachers solved by saying “If my number is 6 times bigger than x/6, then it is 6 times bigger than 5/3”. Another way offered was to analyze the ratio between each numerators and apply it to denominators which had, in order to maintain the equality, to be of the same ratio: “If 6 is the double of 3, then x is the double of 5 which is worth 10”.
This meaning for solving the equation is oriented toward obtaining other equivalent equations to the first one offered, in order to advance toward an equation of the form “x=something”. An example of such was done when solving , where some future teachers doubled the equation, obtaining , which was simpler to read and then multiplied by 5/4 to arrive at x=5/4.
This is related to Arcavi’s (1994) notion of knowing that through transforming an algebraic expression to an equivalent one, it becomes possible to “read” information that was not visible in the original expression. Through these transformations, the intention is not directly to isolate x, but to find other equations, easier ones to read or make sense of, in order to find the value of x.
Albeit treated separately, these varied meanings are not all different and some share attributes. Therefore, in addition to the variety of meanings, significant links can be traced between those, links that can deepen understandings about algebraic equation solving; again, in view of enriching mathematical experiences. For example, meanings 2 and 3 share an explicit orientation toward isolating x, where others do not have this salient preoccupation and focus on other aspects (satisfying the equality, finding points of intersection, etc.). Meanings 4 and 5 share a function orientation in their way of treating the equation, emphasizing each part of the equation as representing an image.
Many meanings also focus implicitly on conditional orientations, be it concerning the satisfaction of the equality or simply the possibility of finding a value for x. For example, in meaning 2 and 3, it is possible that no value of x is found and the same can be said for meaning 4, where it is possible that there be no point of intersection of the two equations or for meaning 5, where possibly no value of x could nullify the left side of the equation (e.g.). Without being explicit about it, these orientations represent a quest for finding a possible value, a quest that can also be unsuccessful. This contrasts heavily with meaning 6, because treating the equation as a ratio assumes or implies that a value of x exists. Meanings 1 and 6 however do share something in common, which is related to an examination of relations between the algebraic unknown and the numbers in order to deduce the value of the algebraic unknown. Both do not opt for a sequence of steps to undertake, but mainly for working with the equation as a whole (in global reading for meaning 1, in ratios for meaning 6). Meanings 3 and 7 share the fact that operations are conducted on the equation as a whole, be it through affecting both sides in the same way to keep the “balance” intact or to obtain new equivalent equations.
Finally, meanings 1, 2, 5 and 6 share the fact that they explicitly look for a number, where the algebraic unknown is conceived as an unknown number that needs to be found; a significant issue to understand when solving algebraic equations (Bednarz & Janvier, 1992; Davis, 1975). Hence, be it through looking at which number could satisfy the equation (meaning 1), which number could nullify a part of it (meaning 5), which number satisfies the proportion (meaning 6) or which is the number on which operations were conducted (meaning 2), all of them focus on x as being a number.
This mental algebra activity, through the variety of strategies but mostly of meanings that emerges, shows promise for algebra teaching and learning. These emerging meanings are significant, because they offer different entry paths into the tasks of solving algebraic equations and do not restrict to a single view of how this can be done. As well, this variety of meanings for what solving an algebraic equation can be offers significant reinvestment opportunities for pushing further the understanding of algebraic equation solving in mathematics teaching. Issues of conditional equations, of deconstructing an equation regarding operations done on a number, of maintaining the balance, of finding equivalent equations, of seeing an equation as a system of equations, and so forth, offered varied ways of conceiving an equation and of solving it. It opened various paths of understanding.
I have not offered details about the discussions that ensued each sharing of strategies, those being contextual and solvers’ understanding related. But, suffice to say that discussions about and explanations of the strategies took an important part of the session: only about 6 tasks were given/solved in 2 hours! Hence, most of the work in the session revolved around explaining, justifying, contrasting and exploring the strength, meaning and relevance of each strategy developed to solve the equations.
In addition, it is through the mental algebra activity that all this emerged, and not through the explicit teaching of these strategies: strategies and meanings became relevant in the need for solving the equations and these meanings were directly connected to those equations. This makes the activity more about the exploration of mathematical ways of meaning, and less about the teaching of explicit strategies for solving. Of course, outcomes will probably vary from one group to other, sometimes offering more, sometimes less. But, in all, it is in the practice of finding ways of solving that these meanings emerged, and their discussion, and it is through this practice that mathematical experiences were enriched and enlarged in relation to ways of solving algebraic equations.
Obviously, this is just one example, and much more is to be explored along these lines with (future) secondary mathematics teachers (and with other mathematical topics!). However, already this emerging variety of meanings through mental algebra shows important promise for enriching algebraic experiences of future teachers, and possibly their students!
Figure 1. Mental process of preservice teachers
1Arcavi gives the example of (2x+3)/(4x+6)=2, which has no solution because whatever the value of x, the numerator is worth half the denominator, making futile undergoing additional steps.
2 This is an avenue also reminiscent of arithmetical divisions, where equivalences are established: e.g. 5.08÷2.54 is equivalent to 508÷254, because 254 divides into 508 the same number of times as 2.54 into 5.08.
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Filloy, E. & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19-25.
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University de Québec